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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 5160–5165
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Spectral elementary-coherence-function representation for partially coherent light pulses

Ari T. Friberg, Hanna Lajunen, and Víctor Torres-Company  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 5160-5165 (2007)
http://dx.doi.org/10.1364/OE.15.005160


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Abstract

A broad class of partially coherent non-stationary fields can be expressed in terms of the recently proposed independent-elementary-pulse model. In this work we first introduce a corresponding dual representation in the frequency domain and then extend this concept by considering shifted and weighted elementary spectral coherence functions. We prove that this method, which closely describes practical optical systems, leads to properly defined correlation functions. As an example, we demonstrate that our new model characterizes, in a natural way, trains of ultra-short pulses, affected by noise and timing jitter, emitted by usual modulators employed in telecom applications.

© 2007 Optical Society of America

1. Introduction

Taking guidance from its spatial counterpart, the notion of Gaussian Schell-model pulse was some time ago introduced to account for possible spectral correlations in pulses whose light intensity profile is a Gaussian [10

10. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002). [CrossRef]

]. Very recently, a new representation was proposed for partially coherent, non-stationary fields in terms of an incoherent superposition of temporally delayed and properly weighted coherent replicas of an elementary source pulse [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]

], in analogy with the corresponding spatial model [12

12. P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006). [CrossRef] [PubMed]

]. The goal of that mathematical model was two-fold: First, it gives a simple and rigorous way to construct a wide variety of arbitrary correlation functions of pulses, and second, it appears to be practically useful, since many light-wave devices operate in this regime [13

13. J. Capmany “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24, 201–229 (2006). [CrossRef]

].

In this work we put forward a different model that can be considered as a generalization of the independent-elementary-pulse representation mentioned above. We first introduce the dual spectral representation, i.e., we consider an incoherent superposition, in the frequency domain, of spectrally shifted and weighted replicas of an elementary spectrally coherent pulse. The advantage of the spectral domain model is that it is closely associated with practical optical situations. In this way, for instance, we recover the well-known effects of finite source line width on stationary light sources that are externally modulated. The spectral representation also provides a basis for the understanding of how temporal partial coherence distorts the ideal comb structure expected from a fully coherent pulse train [7

7. V. Torres-Company, H. Lajunen, and A. T. Friberg, “Effects of partial coherence on frequency combs,” J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1–4 (2007). [CrossRef]

]. We then extend the model by considering, instead of an elementary coherent pulse, an elementary set of spectrally random fields. We demonstrate that this model is consistent with the non-negative definiteness condition. Finally, we show that our new model appears in a natural way when assessing the stochastic variations, such as timing jitter or pulse period fluctuations, of external modulators.

2. Correlation functions in terms of elementary coherent pulses

Partially coherent plane-wave pulses can be studied through coherence functions that measure the correlations of the field at two instants of time or at two frequencies. Simple mathematical models, such as the concept of Gaussian Schell-model pulses [10

10. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002). [CrossRef]

], are useful in assessing the implications of partial coherence, but in reality the correlation functions of non-stationary fields may also take on various complicated forms. In this section we first briefly recall the recently suggested method for constructing correlation functions by a superposition of uncorrelated elementary pulses in the time domain [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]

]. After that, we introduce an analogous representation in the frequency domain and discuss its physical interpretation.

2.1. Time-domain approach

Let us introduce an elementary pulse as a temporally fully coherent complex field, Ue(t) = a(t)exp(-ot), where ωo is the carrier frequency and a(t) represents the complex field envelope. If the realizations U(t) of a partially coherent, random field are assumed to consist of a continuum of such identical, but uncorrelated, elementary pulses, the mutual coherence function can be expressed in the form [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]

]

Γ(t1,t2)=U*(t1)U(t2)=g(t´)Γe(t1t´,t2t´)dt´,
(1)

where the asterisk denotes complex conjugation, Γe(t 1,t 2) = a *(t 1)a(t 2)exp[iω0(t 1t 2)] is the mutual coherence function of the individual deterministic elementary pulses, and g(t) is a (normalized) real and positive weighting function. The angle brackets in the general definition of the mutual coherence function, shown in Eq. (1), denote an ensemble average over the field realizations. Considering the same pulses in the frequency domain, it can be shown that the corresponding cross-spectral density function takes on the form [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]

]

W(ω1,ω2)=A*(ω1ω0)A(ω2ω0)G(ω2ω1),
(2)

where A(ω) and G(ω) are the Fourier transforms of the functions a(t) and g(t), respectively.

It follows from Eq. (2) that the above representation is valid in all cases in which the cross-spectral density function is of the Schell-model type, i.e., the modulus of the spectral degree of coherence depends on Δω= ω2 −ω1 only. As discussed in Ref. [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]

], this model provides an intuitive description for radiation phenomena in terms of classical coherence theory.

2.2. Frequency-domain approach

In the following, we introduce an alternative, dual representation, starting from the frequency domain, for correlation functions in terms of elementary pulses. Let us begin by considering the cross-spectral density function W e12) = Ũ * e1)Ũ e2), where Ũ e(ω) = A(ω−ω0) denotes the Fourier transform of U e(t). Note that since the complex field is taken coherent, the cross-spectral density is a separable function. Now, a broad class of non-stationary coherence functions can be created by spectrally shifting and weighting the above expression with an arbitrary real and positive function WN(ω),

W(ω1,ω2)=WN(ω´)We(ω1ω´,ω2ω´)´.
(3)

This model corresponds to a spectrally incoherent superposition of coherent fields Ũ e(ω). In this way, the energy spectrum of the non-stationary field [14

14. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004). [CrossRef] [PubMed]

], S(ω), can readily be obtained as S(ω) = W(ω, ω), viz.,

S(ω)=WN(ω)U˜e(ω)2;
(4)

in other words, it is a convolution between the weight function WN(ω) and the energy spectrum of the coherent pulse. Of course, the spectral degree of coherence can be obtained at once with the definition

μ(ω1,ω2)=W(ω1,ω2)S(ω1)S(ω2),
(5)

which satisfies 0 ≤ ∣μ12)∣ = 1.

The time-domain coherence function in our model is obtained by recalling the generalized Wiener-Khintchine theorem, i.e.,

Γ(t1,t2)=W(ω1,ω2)exp[i(ω1t1ω2t2)]12.
(6)

Hence, on substituting from Eq. (3) and after some calculations we find that

Γ(t1,t2)=a*(t1)a(t2)exp[iω0(t2t1)]ΓN(t2t1)=Γe(t1,t2)ΓN(t2t1),
(7)

where

ΓN(t2t1)=WN(ω)exp[(t2t1)].
(8)

The average intensity, by definition, then is I(t) = Γ(t,t) = ΓN(0)∣a(t)∣2; thus the temporal profile of the waveform is determined solely by the complex envelope of the elementary pulses. We can also directly see that the degree of coherence, defined as the absolute value of

γ(t1,t2)=Γ(t1,t2)I(t1)I(t2),
(9)

As a practical example of the cross-spectral density functions defined by Eq. (3), we may consider a stationary light source of a mutual coherence function ΓN(t 2t 1) that is externally modulated according to a deterministic temporal format a(t). Such a situation appears frequently, for example, in telecom devices [2

2. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, NY, 2002). [CrossRef]

]. In this case, the function WN(ω) corresponds to the spectral density (or spectrum) of the light source. In particular, when both the modulation format and the spectral density are Gaussian, Gaussian Schell-model pulses are obtained [5

5. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003). [CrossRef] [PubMed]

].

3. Spectral elementary-coherence-function representation

In this section we extend the above model by considering elementary sources that, instead of fully coherent, are partially coherent in the spectral domain. To this end we define an elementary spectral coherence function as

Wpc(ω1,ω2)=U˜pc*(ω1)U˜pc(ω2),
(10)

where Ũ pc(ω) is a field realization of a non-stationary ensemble that is described by the cross-spectral density function W pc12) [15

15. M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” JEOS A: Pure Appl. Opt. 6, 153 (1997). [CrossRef]

]. In the time domain the ensemble can be thought of as being constituted of partially coherent fields given by U pc(t) = a pc(t)exp(−iω0 t), where a pc(t) is a random complex field envelope. In the case of pulses, each field realization can be taken to contain a finite energy and thus be square-integrable. Now, using Eq. (10), let us rewrite Eq. (3) as

W(ω1,ω2)=WN(ω´)Wpc(ω1ω´,ω2ω´)´,
(11)

which gives us the total cross-spectral density in terms of a weighted superposition of shifted elementary coherence functions. The energy spectrum is still found from the convolution S(ω) = WN(ω)⊗S pc(ω), where now S pc(ω) = W pc(ω,ω). It can be shown that the spectral coherence function obtained by Eq. (11) satisfies the non-negative definiteness condition (see Appendix).

In the time domain, recalling Eq. (6), we find that the mutual coherence function in this case is expressed as

Γ(t1,t2)=ΓN(t2t1)Γpc(t1,t2),
(12)

Γpc(t1,t2)=apc*(t1)apc(t2)exp[iω0(t2t1)].
(13)

4. Practical example

As an example of fields described by the elementary-coherence-functions representation, let us now assume that we have a source in which CW laser light is externally modulated. In any realistic situation, the laser is affected by random phase and amplitude changes. Even more, the modulator launches optical pulses with pulse-period fluctuations. In this way, the complex field of the output pulse train can be written as

U(t)=exp(iω0t)N(t)M[tTj(t)],
(14)

where ω0 is the carrier frequency of the laser source and N(t) is a random function that describes its amplitude and phase fluctuations. Further, M(t) = ∑n ψ(t - nT), where ψ(t) denotes the deterministic complex envelope of the pulses and T is the mean period, and j(t) is a real, dimensionless random function of zero mean that describes the timing fluctuations. Now, if the pulse-period fluctuations are small compared with the mean period, we may make a first-order Taylor approximation for the complex field, M[t 1Tj(t)] ≈ M(t) − Tj(t)M(t), where the dot means the time derivative.

Going one step further, we may calculate the mutual coherence function of the global source (laser plus modulator), Γ(t 1,t 2) = ⟨U *(t 1)U(t 2)⟩, by assuming that the noise term N(t) is stationary and taking into account that, since the jitter and the noise originate from different devices, the random processes N(t) and j(t) can be taken as statistically independent. In this way, we find that [8

8. V. Torres-Company, H. Lajunen, and A. T. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B (in press).

]

Γ(t1,t2)=ΓN(t2t1)[M*(t1)M(t2)+T2M˙*(t1)M˙(t2)Γ2(t1,t2)]exp[0(t2t1)],
(15)

with ΓN(t 2t 1) = 〈N*(t 1)N(t 2)〉 and Γj(t 1,t 2) = 〈j(t 1)j(t 2)). The point to emphasize now is that this equation perfectly matches the structure of Eq. (12), by identifying

Γpc(t1,t2)=exp[0(t2t1)][M*(t1)M(t2)+T2M˙*(t1)M˙(t2)Γj(t1,t2)].
(16)

Consequently, the (elementary) cross-spectral density function is determined by Eq. (11), with

Wpc(ω1,ω2)=M˜*(Ω1)M˜(Ω2)+T2Ω1Ω2M˜*(Ω1)M˜(Ω2)2Wj(Ω1,Ω2),
(17)

where Ωi = ωi − ω0 (i = 1,2), ⊗2 denotes two-dimensional convolution, M͂(ω) is the Fourier transform of M(t), and Wj is obtained from Γj by the inverse of Eq. (6).

So, in this practical example, the resulting cross-spectral density function is given by an incoherent superposition of shifted and weighted elementary spectral coherence functions. The weight function is real and positive and corresponds to the power spectrum of the CW laser light, WN(ω) = ∫ΓN(τ)exp[−i(ω −ω0)τ]dτ. On the other hand, if we consider the case with no jitter, i.e., j(t) = 0, the elementary coherence functions become fully coherent, corresponding the situations of Section 2.2.

5. Summary and conclusions

We have introduced novel mathematical models for the coherence functions of non-stationary plane-wave fields and pointed out how they appear in practical physical situations dealing with random pulses and pulse trains, First, the space–frequency domain analogue of the recent independent-elementary-pulse representation was discussed. The concept was then extended by considering a new representation based on a weighted superposition of shifted elementary spectral coherence functions. It was further shown that the resulting representation satisfies the sufficiency condition required of correlation functions. Several examples of actual physical situations that correspond to these new frequency domain models were briefly discussed.

Finally, a corresponding elementary-coherence-function representation could naturally also be introduced starting from the space–time domain. As with the models dealing with coherent elementary pulses, the two dual approaches together could cover a wider class a partially coherent wave fields. However, as was demonstrated by the examples above, a clear physical association can be found for the frequency domain representation in many cases. Hence, in our opinion, it corresponds to the representation that is more closely anchored to practical experimental arrangements.

Appendix A: Proof of the non-negative definiteness condition

In order to be a proper correlation function, the cross-spectral density defined by Eq. (11) must satisfy the non-negative definiteness condition

f*(ω1)f(ω2)W(ω1,ω2)120,
(A1)

where f(ω) is an arbitrary, sufficiently well-behaved function [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

]. Based on Eqs. (10) and (11) we find that

f*(ω1)f(ω2)Wpc(ω1ω´,ω2ω′)WN(ω′)12dω′
=f*(ω1)U˜pc*(ω1ω´)1f(ω2)U˜pc(ω2ω´)2WN(ω´)´
=f(ω´)U˜pc(ω´)2WN(ω´)´0,
(A2)

where, in the second step, we have changed the order of integration and ensemble averaging, and the last inequality follows from the fact that WN(ω) is a real and positive function. Thus any function obtained by the representation of Eq. (11) is indeed a proper correlation function. Furthermore, this proof is valid also for the independent-elementary-pulse representation discussed in section 2, since that constitutes a special case (of coherent elementary fields) of the more general approach.

Acknowledgments

V. Torres is grateful for funding from the Dirección General de Investigación Cientifica y Técnica, Spain, and FEDER, under project FIS2004-02404. He also acknowledges financial support from a FPU grant of the Ministerio de Educación y Ciencia, Spain. H. Lajunen thanks the Academy of Finland (project 111701), the Network of Excellence on Micro-Optics (NEMO, http://www.micro-optics.org), the Helsingin Sanomat Centennial Foundation, and the Vilho, Yrjö, and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. A. T. Friberg acknowledges funding from the Swedish Foundation for Strategic Research (SSF).

References and links

1.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

2.

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, NY, 2002). [CrossRef]

3.

B. E. A. Saleh and M. I. Irshid, “Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber,” Opt. Lett. 7, 342–343 (1982). [CrossRef] [PubMed]

4.

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30, 2973–2975 (2005). [CrossRef] [PubMed]

5.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003). [CrossRef] [PubMed]

6.

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Noise-induced linewidth in frequency combs,” Opt. Lett. 31, 1875–1877 (2006). [CrossRef] [PubMed]

7.

V. Torres-Company, H. Lajunen, and A. T. Friberg, “Effects of partial coherence on frequency combs,” J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1–4 (2007). [CrossRef]

8.

V. Torres-Company, H. Lajunen, and A. T. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B (in press).

9.

S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003). [CrossRef]

10.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002). [CrossRef]

11.

P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]

12.

P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006). [CrossRef] [PubMed]

13.

J. Capmany “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24, 201–229 (2006). [CrossRef]

14.

S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004). [CrossRef] [PubMed]

15.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” JEOS A: Pure Appl. Opt. 6, 153 (1997). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(320.5550) Ultrafast optics : Pulses

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: February 7, 2007
Revised Manuscript: April 12, 2007
Manuscript Accepted: April 12, 2007
Published: April 13, 2007

Citation
Ari T. Friberg, Hanna Lajunen, and Victor Torres-Company, "Spectral elementary-coherence-function representation for partially coherent light pulses," Opt. Express 15, 5160-5165 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5160


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References

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  2. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, NY, 2002). [CrossRef]
  3. B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 342-343 (1982). [CrossRef] [PubMed]
  4. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent planewave- type pulses," Opt. Lett. 30, 2973-2975 (2005). [CrossRef] [PubMed]
  5. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003). [CrossRef] [PubMed]
  6. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875- 1877 (2006). [CrossRef] [PubMed]
  7. V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007). [CrossRef]
  8. V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).
  9. S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003). [CrossRef]
  10. P. P¨a¨akk¨onen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002). [CrossRef]
  11. P. Vahimaa and J. Turunen, "Independent-elementary-pulse representation for non-stationary fields," Opt. Express 14, 5007-5012 (2006). [CrossRef] [PubMed]
  12. P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14, 1376-1381 (2006). [CrossRef] [PubMed]
  13. J. Capmany "A tutorial on microwave photonic filters," J. Lightwave Technol. 24, 201-229 (2006). [CrossRef]
  14. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004). [CrossRef] [PubMed]
  15. M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997). [CrossRef]

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